Dynamic Adaptive Resource Scheduling for Phased Array Radar: Enhancing Efficiency through Synthesis Priorities and Pulse Interleaving

Most adaptive scheduling algorithms focus primarily on the time resources of the radar system, often overlooking constraints related to energy and computational resources. While it can be argued that the bottleneck posed by computational resource limitations is gradually diminishing due to advancements in computer performance, the energy resource constraints of the radar system remain critical, particularly as radar cannot remain in a transmitting state for extended periods. The implementation of a pulse interleaving strategy allows the waiting periods between radar pulses to be utilized for transmitting other radar pulses.

Tracking tasks can be classified according to the threat levels associated with operational conditions, which may include high-precision tracking, precision tracking, ordinary tracking, or simple monitoring. Figure 2 illustrates the task model of pulsed phased array radar, which consists of three components: the launch period, waiting period, and reception period. The i-th phased array radar task can be described as follows:

In  (1), $P_{ri}$ represents the task type, $t_{ei},\eta_{i},t_{si}$ stand for the task request time, task attribute, and actual execution time respectively. $\Delta t_{i}$ represents the beam dwell time of the task. $l_{i}$ is the time window length, $t_{xi}$ for the transmit interval, $t_{wi}$ for the wait interval, which depends on the target distance, and $t_{ri}$ for the receive interval. $\Delta t_{i}$ satisfies:

where $t_{ci}$ is the pulse launch cooling time, and $t_{di}$ is the deadline. Each task should be executed before its deadline, $t_{di}$ satisfies:

$\Delta T_{i}$ is the sample interval between two adjacent tasks of the same type. It satisfies:

where $t_{s(i-1)}$ denotes the execution time of the former task. $P_{ti}$ is the task power consumption. In reality, the transmit interval $t_{xi}$ consumes the majority of the power of the whole task.

As the task scheduling of each working mode needs to be realized through the radar pulses, it will consume certain system energy and time resources. Considering that the radar cannot always be in the transmitting state, energy resource constraints should be applied to the phased array radar when the pulse interlace strategy is used.

In order to maximize resource utilization and scheduling efficiency, various resource constraints must be considered when performing resource scheduling.

1) Energy Resource Constraint

Radar systems are well-known to consume a significant amount of energy during pulse transmission. When pulse streams are interlaced, the operational time of the radar transmitter is extended, leading to increased energy consumption. If the operating temperature of the radar exceeds the physical limits of the system, there is a risk of damaging the transmitter. During the tracking process, to maintain adequate tracking quality, the width of the transmitted pulse must be increased, which further escalates energy consumption. It has been observed that energy constraints are influenced by the selected modes of operation, which impacts scheduling analysis in the following ways:

In (2), $P_{\tau_{\text{max}}}$ is the upper limit of radar instantaneous power consumption. $P_{\tau}(t)$ is the power consumed by the radar transmitting a pulse at time $t$.

The cooling time constant is $\tau$, which characterizes the heat dissipation performance of the radar. The energy consumed during a radar receiving period is negligible, and the time difference between the end of a pulse emission period and the start time of the next pulse transmission period is $t_{ci}$. It should be ensured that the energy at the end of the next scheduled mission launch period does not exceed the instantaneous maximum power.

2) Time Resources Constraint

Because radar scheduling tasks are mainly performed at regular scheduling intervals, once the scheduling interval is determined, the radar task scheduling becomes less flexible. The launch and reception periods of radar missions cannot be preempted, and all missions must be completed before their deadlines. In a traditional scheduling algorithm, a single beam dwell period cannot be preempted, and the number of tasks, $N_{1}$, successfully executed within one scheduling interval must satisfy:

In (5), $\Delta t_{k}$ is the length of the beam dwell period, and $t_{sI}$ is the duration of the scheduling interval. In order to improve the utilization of radar time resources, the pulse interleaving strategy is employed to interlace other tasks during the waiting period between each radar task reception and transmission. The pulse transmission period and reception period of a task cannot be preempted by other tasks; otherwise, the task execution will fail. Successful execution of $N_{i}$ tasks imposes the following constraints:

In the task priority design process, this paper proposes an approach that first considers the task working mode, which reflects the importance of the task, and the task deadline, which indicates the urgency of the task. A priority table is designed to evaluate the importance and urgency of each task simultaneously, assigning them equivalent weights. Building upon this framework, a time deviation ratio is incorporated to enhance the efficiency of task scheduling, ensuring that critical tasks are executed as promptly as possible within their expected execution times. Ultimately, a synthesis priority design method is developed that accounts for task working modes, deadlines, and time shift ratios.

As shown in 3, the $k$-th tracking target position is $Z(k)$, the predicted value of the $k+1$-th tracking target position is $\hat{Z}(k+1/k)$, the target measurement position is $Z(k+1)$, and the new information is $v(k+1)$. When there is a time offset, the target measurement position also changes; the new measurement position of the target will be recorded as $Z_{1}(k+1)$, and the new interest is $v_{1}(k+1)$. Consequently, the target tracking error will increase, and the new information will grow to a certain extent. Meanwhile, the target will fall into the tracking wave door with a greater probability, which may lead to tracking failure. Since the target tracking is filtered by a tracking algorithm such as an extended Kalman filter, the effectiveness of the target tracking is a quadratic function of the time offset [13].

Therefore, in the comprehensive task priority design, the time shift should be synthesized into the task priority in the form of a quadratic function, maximizing the benefits of task scheduling and improving the effectiveness of task scheduling.

It is necessary to clarify the basic principles of task scheduling: (1) the higher the priority of the work mode, the higher the priority of the task; (2) the earlier the deadline, the higher the priority of the task; (3) the lower the time shift ratio, the higher the priority of the task.

Step 1: The overall priority of the task to be scheduled is

Suppose that the current time scheduler has $Q$ requests, which are recorded as $q=\{q_{1},q_{2},\ldots,q_{Q}\}$, satisfying: (1) the arrival time of all requests is not greater than the current time; (2) the deadline of all requests minus the residence time is not less than the current time.

First, the priority in accordance with the mode of operation of the size of the cut-off request $Q$ tasks are arranged twice. Each task number is obtained in the arrangement, referred to as priority number $N_{p_{i}}$, and the deadline $N_{d_{i}}$, where $1\leq N_{p_{i}},N_{d_{i}}\leq Q$. Then,

where $\eta$ is a weighted value in the range of $[1,Q]$.

Step 2: The accuracy of tracking targets has a quadratic function relationship with the time shift. In order to make each scheduling task as close to its expected execution time as possible, the closer the time pointer is to the expected execution time, the higher the task priority is.

If the current request scheduling task operates in one of three modes: validation, precision tracking, and normal tracking, then

If the current working mode of the request scheduling task is airspace search or horizontal search, then

In Eq.(11) and Eq.(12), the time pointer $tp$ points to the current time $\alpha\in(0,1),b\in(0,1)$. As the time window of validation and task tracking is smaller than that of task searching, the priority of validation and task tracking is higher than that of task searching under the same time shift ratio.

This section proposes a method to evaluate the performance of the proposed scheduling algorithm:

Successful Scheduling Ratio

The Successful Scheduling Ratio (SSR) is defined as the ratio between the number of successfully scheduled tasks and the total number of request tasks. It can be expressed as:

In (13), $N_{1}$ is the number of successfully scheduled tasks, and $M$ is the total number of request tasks. The more tasks are successfully scheduled, the better the performance of the algorithm is.

Time Utilization Ratio

The Time Utilization Ratio (TUR) is the ratio of the launch duration and reception duration of all successfully scheduled tasks to the total time resource, which can be expressed as:

where $T_{0}$ is the total available time resource, and $N$ is the number of successfully scheduled tasks. The higher the time utilization ratio in a phased array radar, the better the scheduling algorithm.

Average Time Shift Ratio

The Average Time Shift Ratio (ATSR) is the mean of the difference between the actual execution time of the successful scheduling tasks and the ideal execution time of them. This metric evaluates the effectiveness of task scheduling.

where $N$ is the number of scheduled successful tasks, $t_{si}$ is the actual execution time, $t_{ei}$ is the task execution time, and $l_{i}$ is the time window length of the task.

Scheduling Yield Ratio

The Scheduling Yield Ratio (SYR) is the ratio of the total yield of all tasks successfully scheduled to the ideal yield of all tasks. Since most classical Kalman filtering algorithms are nonlinear filtering algorithms, the target tracking accuracy is nonlinear with the time shift, and consequently, the effectiveness of the radar task scheduling is expressed as:

Then the Scheduling Yield Ratio can be expressed as:

The importance of the various task types is expressed by $k$, which could be $1/2$.

To maximize the utilization of time resources and enhance the success ratio of radar task scheduling, it is essential to implement a pulse interleaving strategy. Building upon the dynamic synthesis priority model proposed in this paper, we develop a dynamic pulse interleaving algorithm designed to mitigate the complexity introduced by the integration of pulse interleaving.

Figure 4 shows two pulse interleaving modes for tracking tasks. Figure 4 (a) illustrates an external interleaving type pulse interleaving mode, while Figure 4 (b) depicts an internal interleaving type pulse interleaving mode. The blank rectangle represents the radar task 1, and the black rectangle represents the radar task 2. $t_{1},t_{2}$ are the cooling times of the radar transmitter. The $o_{1}$ indicates the internal idle time of the pulse after pulse interleaving.

The design principle of pulse interleaving between task 1 and task 2 is as follows: for impulse interleaving mode 1, minimize $w_{1}$ as much as possible; for pulse interleaving mode 2, minimize $w_{2}$ as much as possible, such that the reception of the adjacent radar pulses reception periods $t_{r1}$ and $t_{r2}$ is as close as possible.

The time must satisfy the following equations:

Assuming that there are $N$ tasks waiting to be scheduled at time period $[t_{s(1)},t_{end}]$, the specific steps of pulse interleaving are as follows:

Step 1: Initialize the time pointer $tp=t_{0}$, the number of tasks that are scheduled $q=0$, radar pulse transmit power $A_{1}$, previous radar pulse cooldown time $t_{c}=0$, radar pulse launch period end time power $P_{out}=0$, and pulse internal idle available time after pulse interleaving is $o_{1}=0$, and the pulse external idle available time is $o_{2}=[t_{s(1)},t_{end}]$.

Step 2: Select all the tasks that satisfy $tp\subset[t_{ei}-l_{i},t_{ei}+l_{i}-\Delta t_{i}]$ and dynamically apply the synthesis priority design method according to (9) to obtain a dynamic priority for each task and select the task with the highest priority.

Step 3: Determine whether the task to be scheduled can be pulse-interlaced with the task from the previous moment.

Case (1): If pulse interleaving is possible, the parameters after updating the interlaced task are as follows:

Case (2): If the task to be scheduled is a tracking task and cannot perform pulse interleaving, update the parameters as follows:

Case (3): If the task to be scheduled is different and it cannot interlace, the parameters are updated as follows:

Case (4): If there is no task request execution at the current time, update the time pointer $tp=tp+dt$, where $dt$ is the minimum moving step size.

Step 4: When $tp>t_{end}$ or $q>N$, go to Step 6; otherwise, return to Step 2.

Step 5: Delete the task that is not scheduled for execution, obtain the schedule execution linked list, delete the linked list, and end the task scheduling analysis in the scheduling interval.

Fig. 4 shows the algorithm scheduling process. When performing task scheduling, it satisfies the time constraints in [eq:18], $P_{out}\leq P_{max}$ energy constraint, and the dynamic scheduling update of allocation scheduling tasks on the timeline in the form of time pointers, which simplifies the complexity of scheduling analysis.

The simulation scenario includes five types of tasks to be scheduled: verification, precision tracking, general tracking, horizon search, and airspace search. Task tracking is categorized into precision tracking and general tracking based on the level of target threat. The verification radar task consists of two components: one for generating verification tasks for target detection, and the other for generating false alarm tasks at a specific data ratio. The number of general target tracking tasks is fixed at 40, while the number of precision tracking tasks gradually increases from 0 to 200 in increments of 10. Search tasks are generated according to predefined data ratios.

The radar detection range is assumed to be 50-300 km, with a simulation time of 0-12 seconds. The phased array radar scheduling interval is set to SI = 50 ms, and targets appear randomly. Due to the random generation of targets, the tracking task exhibits a degree of randomness. The number of precision tracking targets increases by 10 with each iteration, and the simulation is repeated 100 times. The parameters for the radar task types are summarized in Table I.

The simulation results are compared with those from previous works, including traditional time-hand-based methods [9], [12], and [21]. A section of the original task request sequence, from 200 ms to 300 ms, is extracted for analysis, as shown in Figure 6.

As shown in Fig. 6 and Fig. 7, the scheduling yield of the dynamic pulse interleaving algorithm is not only higher than that of the traditional time-pointer algorithm presented in [9], but also exceeds the yield of the quadratic programming algorithm in [12]. Additionally, the dynamic pulse interleaving algorithm achieves a higher scheduling yield compared to the pulse interleaving algorithm in Zhang’s work [21].

When the number of precision tracking targets is below 50, the original quadratic programming algorithm achieves the highest yield, outperforming other algorithms, with the pulse interleaving algorithm ranking second. In this scenario, the scheduling gain is greater, and the scheduling failure ratio is the lowest. As the number of precision tracking targets increases, the dynamic pulse interleaving algorithm shows the highest scheduling yield ratio, surpassing both the traditional pulse interleaving algorithm and other methods. Furthermore, the overall scheduling failure ratio of the dynamic pulse interleaving algorithm remains consistently lower than that of other algorithms.

As shown in Fig. 8, the time utilization ratio of the pulse interleaving algorithm proposed in this paper is consistent with that of the method in [21], and it surpasses other algorithms in terms of time utilization. In Fig. 9, it can be observed that the time shift ratio of the proposed method, which incorporates task synthesis priority with time shift considerations, is lower than other algorithms, and only slightly higher than that of the quadratic programming algorithm in [12]. As the number of precision tracking targets increases, the time shift ratio of the proposed method approaches that of [12], further enhancing the effectiveness of task execution.

The pulse interleaving algorithm presented in this paper shows similar performance to [21] in terms of scheduling time utilization and overall failure ratios. However, the time shift ratio is reduced by 35%, and the scheduling gain is improved by 30%. Compared to [9], the time utilization ratio is improved by nearly 20%, the time shift ratio is reduced by 40%, and the scheduling benefit is increased by 45%.

As shown in Fig. 10, the scheduling failure ratios for verification tasks using this method are similar to those of other scheduling algorithms. However, Figs. 11, 12, 13, and 14 demonstrate that the proposed algorithm achieves the lowest scheduling failure ratios across precision tracking, general tracking, high-priority search, and low-priority search tasks. Notably, the failure ratio for general tracking task scheduling is significantly lower than that of other algorithms.